Oscillations exhibited by nonlinear systems of differential equations are qualitatively different when they are forced by an external oscillation applied to the system. Although some solutions of the forced system have, not surprisingly, the same period as the forcing term, there exist other solutions whose period is some integral multiple of the forcing period. Numerical computer techniques were implemented to follow the periodic orbits and to calculate their period and stability. The methods were applied to an epidemic model (SEIR) with birth and death rates added to insure population turnover and a steady supply of new susceptibles. The external oscillation was put into the transmission term, corresponding here to seasonal changes. The major result is that it is relatively easy to find stable periodic solutions whose periods are double that of the forcing term corresponding to some measles epidemics. Solutions with longer periods generally have much greater amplitudes. Consequently, observed non-biennial cycles of epidemics, such as those of rubella, probably are fundamentally different.